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arxiv: 2604.03218 · v1 · submitted 2026-04-03 · 🧮 math.ST · math.PR· stat.ML· stat.TH

Recognition: 2 theorem links

· Lean Theorem

Power one sequential tests exist for weakly compact mathscr P against mathscr P^c

Ashwin Ram , Aaditya Ramdas
Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:30 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.MLstat.TH
keywords sequential testingpower-one testsweak compactnesse-processescomposite hypothesesPolish spacesi.i.d. sampling
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The pith

A level-α sequential test with power one against the complement exists for any weakly compact null set of distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that when the null hypothesis consists of a weakly compact collection of probability measures on a Polish space, it is possible to construct a sequential test that controls the type I error at level α while achieving power one against every distribution outside the null. This means the test will eventually reject the null with probability one if the data comes from any alternative distribution. A sympathetic reader would care because such tests allow for continuous monitoring of data without inflating error rates, a property impossible for fixed-sample tests. The result holds under i.i.d. sampling without additional restrictions on the distributions beyond weak compactness of the null set. It also shows how to combine these tests into an e-process that grows to infinity under alternatives.

Core claim

For i.i.d. observations in Polish spaces, if the set of null distributions P is weakly compact in the weak topology, then there exists a level-α sequential test that has power one against P^c. Such tests can be aggregated to form an e-process for P that tends to infinity under any alternative in P^c, and an asymptotically relatively growth-rate optimal e-process can be constructed.

What carries the argument

Weak compactness of the null set P in the weak topology on probability measures, which enables a covering or aggregation argument to build a sequential test guaranteeing the power-one property under i.i.d. sampling.

If this is right

  • Level-α sequential tests with power one exist for any weakly compact P.
  • These tests aggregate into an e-process for P that increases to infinity under P^c.
  • An asymptotically relatively growth rate optimal e-process against P^c can be constructed.
  • The power-one property holds for i.i.d. data in Polish spaces with no further restrictions beyond weak compactness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • When the null set fails to be weakly compact, power-one tests may still exist in special cases even though the paper's sufficient condition no longer applies.
  • The result implies that parametric families whose closure is weakly compact admit anytime-valid sequential tests for continuous data monitoring.
  • e-process constructions derived from these tests extend naturally to anytime-valid inference for composite null hypotheses.
  • Similar compactness arguments might yield power-one tests in certain dependent or non-i.i.d. settings where an appropriate topology preserves compactness.

Load-bearing premise

The null hypothesis set of distributions is weakly compact under the weak topology.

What would settle it

A weakly compact set P in a Polish space for which every level-α sequential test has power strictly less than one against some distribution in P^c.

read the original abstract

Suppose we observe data from a distribution $P$ and we wish to test the composite null hypothesis that $P\in\mathscr P$ against a composite alternative $P\in \mathscr Q\subseteq \mathscr P^c$. Herbert Robbins and coauthors pointed out around 1970 that, while no batch test can have a level $\alpha\in(0,1)$ and power equal to one, sequential tests can be constructed with this fantastic property. Since then, and especially in the last decade, a plethora of sequential tests have been developed for a wide variety of settings. However, the literature has not yet provided a clean and general answer as to when such power-one sequential tests exist. This paper provides a remarkably general sufficient condition (that we also prove is not necessary). Focusing on i.i.d. laws in Polish spaces without any further restriction, we show that there exists a level-$\alpha$ sequential test for any weakly compact $\mathscr P$, that is power-one against $\mathscr P^c$ (or any subset thereof). We show how to aggregate such tests into an $e$-process for $\mathscr P$ that increases to infinity under $\mathscr P^c$. We conclude by building an $e$-process that is asymptotically relatively growth rate optimal against $\mathscr P^c$, an extremely powerful result.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for i.i.d. observations in Polish spaces, any weakly compact collection P of probability measures admits a level-α sequential test with power one against P^c (or any subset). The argument constructs an e-process that is a supermartingale under every law in P and diverges almost surely under any Q ∉ P, relying on weak compactness and Prohorov tightness for uniform type-I control and separation. It further shows how to aggregate such tests into an e-process for P and constructs one that is asymptotically relatively growth-rate optimal against P^c.

Significance. If the central existence claim holds, the result supplies a remarkably general topological sufficient condition (weak compactness in the weak topology) for power-one sequential tests without further restrictions on P. This generalizes classical work of Robbins et al. and recent e-process literature by providing both existence and explicit constructions, including an optimality result on relative growth rates. The paper ships a clean, non-circular argument resting on standard tightness properties rather than data-dependent or self-referential constructions.

minor comments (2)
  1. [§2] §2: the definition of the e-process supermartingale property could be stated with an explicit reference to the filtration to avoid any ambiguity about the optional stopping used later.
  2. [Remark after Theorem 3.2] The statement that weak compactness is 'not necessary' is mentioned in the abstract but the counter-example or necessity argument appears only in a remark; moving a brief sketch to the main text would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation to accept. We have no revisions to propose in response to this report.

Circularity Check

0 steps flagged

Existence proof is topologically grounded without circularity

full rationale

The paper establishes an existence result: for i.i.d. observations in Polish spaces, any weakly compact null set P admits a level-α sequential test (equivalently, an e-process) that is a supermartingale under every law in P and diverges almost surely under any Q outside P. The derivation invokes weak compactness to obtain Prohorov tightness, which supplies the uniform integrability and separation needed for both type-I control and power one; these are standard facts from measure-theoretic probability and do not reduce to any fitted parameter, self-definition, or load-bearing self-citation chain. No equation or construction is shown to be equivalent to its own inputs by construction, and the argument remains externally verifiable against classical tightness and martingale theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard measure-theoretic assumptions for Polish spaces and i.i.d. sampling together with the paper-specific topological assumption of weak compactness; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Observations are i.i.d. from laws on a Polish space
    Explicitly stated as the setting for the existence result.
  • ad hoc to paper Weak compactness of P is sufficient for existence of power-one tests
    This is the load-bearing condition proved in the paper.

pith-pipeline@v0.9.0 · 5541 in / 1089 out tokens · 42208 ms · 2026-05-13T18:30:06.628879+00:00 · methodology

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Reference graph

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